Question

The matrix A=\(\begin{bmatrix}2&9\\2&6\end{bmatrix}\) as a sum of symmetric and skew-symmetric matrix is ______

(a) \( \frac{1}{4} \begin{bmatrix}4&11\\11&12\end{bmatrix} – \frac{1}{2} \begin{bmatrix}0&7\\-7&0\end{bmatrix}\)

(b) \( \frac{1}{4} \begin{bmatrix}4&11\\11&12\end{bmatrix} + \frac{1}{2} \begin{bmatrix}0&7\\7&0\end{bmatrix}\)

(c) \( \frac{1}{2} \begin{bmatrix}4&11\\11&12\end{bmatrix} + \frac{1}{2} \begin{bmatrix}0&7\\-7&0\end{bmatrix}\)

(d) \( \frac{1}{2} \begin{bmatrix}4&11\\11&12\end{bmatrix} – \frac{1}{2} \begin{bmatrix}0&7\\-7&0\end{bmatrix}\)

I have been asked this question during a job interview.

Question is taken from Symmetric and Skew Symmetric Matrices in division Matrices of Mathematics – Class 12

Answer 1

Correct choice is (c) \( \frac{1}{2} \begin{bmatrix}4&11\\11&12\end{bmatrix} + \frac{1}{2} \begin{bmatrix}0&7\\-7&0\end{bmatrix}\)

The best explanation: Given that A=\(\begin{bmatrix}2&9\\2&6\end{bmatrix}\).

A’=\(\begin{bmatrix}2&2\\9&6\end{bmatrix}\)

⇒A+A’=\(\begin{bmatrix}2&9\\2&6\end{bmatrix}\)+\(\begin{bmatrix}2&2\\9&6\end{bmatrix}\)=\(\begin{bmatrix}4&11\\11&12\end{bmatrix}\)

⇒A-A’=\(\begin{bmatrix}2&9\\2&6\end{bmatrix}\)–\(\begin{bmatrix}2&2\\9&6\end{bmatrix}\)=\(\begin{bmatrix}0&7\\-7&0\end{bmatrix}\)

The given square matrix can be written as

⇒A = \( \frac{1}{2}\) (A+A’) + \( \frac{1}{2}\) (A-A’)=\( \frac{1}{2} \begin{bmatrix}4&11\\11&12\end{bmatrix} + \frac{1}{2} \begin{bmatrix}0&7\\-7&0\end{bmatrix}\).